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Transcript
Text Mining Techniques for Leveraging Positively Labeled Data
Lana Yeganova*, Donald C. Comeau, Won Kim, W. John Wilbur
National Center for Biotechnology Information, NLM, NIH, Bethesda, MD 20894 USA
{yeganova, comeau, wonkim, wilbur}@mail.nih.gov
* Corresponding author. Tel.:+1 301 402 0776
Abstract
Suppose we have a large collection of
documents most of which are unlabeled. Suppose
further that we have a small subset of these
documents which represent a particular class of
documents we are interested in, i.e. these are
labeled as positive examples. We may have reason
to believe that there are more of these positive
class documents in our large unlabeled collection.
What data mining techniques could help us find
these unlabeled positive examples? Here we
examine machine learning strategies designed to
solve this problem. We find that a proper choice of
machine learning method as well as training
strategies can give substantial improvement in
retrieving, from the large collection, data enriched
with positive examples. We illustrate the principles
with a real example consisting of multiword
UMLS phrases among a much larger collection of
phrases from Medline.
1
Introduction
Given a large collection of documents, a few of
which are labeled as interesting, our task is to
identify unlabeled documents that are also
interesting. Since the labeled data represents the
data we are interested in, we will refer to it as the
positive class and to the remainder of the data as
the negative class. We use the term negative class,
however, documents in the negative class are not
necessarily negative, they are simply unlabeled and
the negative class may contain documents relevant
to the topic of interest. Our goal is to retrieve these
unknown relevant documents.
A naïve approach to this problem would simply
take the positive examples as the positive class and
the rest of the collection as the negative class and
apply machine learning to learn the difference and
rank the negative class based on the resulting
scores. It is reasonable to expect that the top of this
ranking would be enriched for the positive class.
But an appropriate choice of methods can improve
over the naïve approach.
One issue of importance would be choosing the
most appropriate machine learning method. Our
problem can be viewed from two different
perspectives: the problem of learning from
imbalanced data as well as the problem of
recommender systems. In terms of learning from
imbalanced data, our positive class is significantly
smaller than the negative, which is the remainder
of the collection. Therefore we are learning from
imbalanced data. Our problem is also a
recommender problem in that based on a few
examples found of interest to a customer we seek
similar positive examples amongst a large
collection of unknown status. Our bias is to use
some form of wide margin classifier for our
problem as such classifiers have given good
performance for both the imbalanced data problem
and the recommender problem (Zhang and Iyengar
2002; Abkani, Kwek et al. 2004; Lewis, Yang et
al. 2004).
Imbalanced data sets arise very frequently in
text classification problems. The issue with
imbalanced learning is that the large prevalence of
negative documents dominates the decision
process and harms classification performance.
Several approaches have been proposed to deal
with the problem including sampling methods and
cost-sensitive learning methods and are described
in (Chawla, Bowyer et al. 2002; Maloof 2003;
Weiss, McCarthy et al. 2007). These studies have
shown that there is no clear advantage of one
approach versus another. Elkan (2001) points out
that cost-sensitive methods and sampling methods
are related in the sense that altering the class
distribution of training data is equivalent to
altering misclassification cost. Based on these
studies we examine cost-sensitive learning in
which the cost on the positive set is increased, as a
useful approach to consider when using an SVM.
In order to show how cost-sensitive learning for
an SVM is formulated, we write the standard
equations for an SVM following (Zhang 2004).
155
Proceedings of the 2011 Workshop on Biomedical Natural Language Processing, ACL-HLT 2011, pages 155–163,
c
Portland, Oregon, USA, June 23-24, 2011. 2011
Association for Computational Linguistics
Given training data
 x , y  where
i
i
yi is 1 or –1
depending on whether the data point xi is
classified as positive or negative, an SVM seeks
that vector wi which minimizes

 h  y (x  w   )  2
i
i
i
w
2
(1)
where the loss function is defined by
 1  z, z  1
h z  
z>1.
 0,
(2)
The cost-sensitive version modifies (1) to become
r   h  yi ( xi  w   )   r   h  yi ( xi  w   )  
iC
iC

2
w
2
(3)
and now we can choose r and r to magnify the
losses appropriately. Generally we take r to be 1,
and r to be some factor larger than 1. We refer to
this formulation as CS-SVM. Generally, the same
algorithms used to minimize (1) can be used to
minimize (3).
Recommender systems use historical data on
user preferences, purchases and other available
data to predict items of interest to a user. Zhang
and Iyengar (2002) propose a wide margin
classifier with a quadratic loss function as very
effective for this purpose (see appendix). It is used
in (1) and requires no adjustment in cost between
positive and negative examples. It is proposed as a
better method than varying costs because it does
not require searching for the optimal cost
relationship between positive and negative
examples. We will use for our wide margin
classifier the modified Huber loss function (Zhang
2004). The modified Huber loss function is
quadratic where this is important and has the form
4  z, z  1

2
h  z   1  z  , -1  z  1
(4)

0, z>1.

We also use it in (1). We refer to this approach as
the Huber method (Zhang 2004) as opposed to
SVM. We compare it with SVM and CS-SVM. We
used our own implementations for SVM, CS-SVM,
and Huber that use gradient descent to optimize the
objective function.
156
The methods we develop are related to semisupervised learning approaches (Blum and
Mitchell 1998; Nigam, McCallum et al. 1999) and
active learning (Roy and McCallum 2001; Tong
and Koller 2001). Our method differs from active
learning in that active learning seeks those
unlabeled examples for which labels prove most
informative in improving the classifier. Typically
these examples are the most uncertain. Some semisupervised learning approaches start with labeled
examples and iteratively seek unlabeled examples
closest to already labeled data and impute the
known label to the nearby unlabeled examples. Our
goal is simply to retrieve plausible members for the
positive class with as high a precision as possible.
Our method has value even in cases where human
review of retrieved examples is necessary. The
imbalanced nature of the data and the presence of
positives in the negative class make this a
challenging problem.
In Section 2 we discuss additional strategies
proposed in this work, describe the data used and
design of experiments, and provide the evaluation
measure used. In Section 3 we present our results,
in Sections 4 and 5 we discuss our approach and
draw conclusions.
2
Methods
2.1
Cross Training
Let D represent our set of documents, and C
those documents that are known positives in D .
Generally C would be a small fraction of D and
for the purposes of learning we assume that
C  D \ C .
We are interested in the case when some of the
negatively labeled documents actually belong to
the positive class. We will apply machine learning
to learn the difference between the documents in
the class C and documents in the class C and
use the weights obtained by training to score the
documents in the negative class C . The highest
scoring documents in set C are candidate
mislabeled documents. However, there may be a
problem with this approach, because the classifier
is based on partially mislabeled data. Candidate
mislabeled documents are part of the C class. In
the process of training, the algorithm purposely
learns to score them low. This effect can be
magnified by any overtraining that takes place. It
will also be promoted by a large number of
features, which makes it more likely that any
positive point in the negative class is in some
aspect different from any member of C .
Another way to set up the learning is by
excluding documents from directly participating in
the training used to score them. We first divide the
negative set into disjoint pieces
C  Z1  Z 2
Then train documents in C versus documents in
Z1 to rank documents in Z 2 and train documents
in C versus documents in Z 2 to rank documents
in Z 1 . We refer to this method as cross training
(CT). We will apply this approach and show that it
confers benefit in ranking the false negatives in
C .
2.2
Data Sources and Preparation
The databases we studied are MeSH25, Reuters,
20NewsGroups, and MedPhrase.
MeSH25. We selected 25 MeSH® terms with
occurrences covering a wide frequency range: from
1,000 to 100,000 articles. A detailed explanation of
MeSH
can
be
found
at
http://www.nlm.nih.gov/mesh/.
For a given MeSH term m, we treat the records
assigned that MeSH term m as positive. The
remaining MEDLINE® records do not have m
assigned as a MeSH and are treated as negative.
Any given MeSH term generally appears in a small
minority of the approximately 20 million MEDLINE
documents making the data highly imbalanced for
all MeSH terms.
Reuters. The data set consists of 21,578 Reuters
newswire articles in 135 overlapping topic
categories. We experimented on the 23 most
populated classes.
For each of these 23 classes, the articles in the
class of interest are positive, and the rest of 21,578
articles are negative. The most populous positive
class contains 3,987 records, and the least
populous class contains 112 records.
157
20NewsGroups. The dataset is a collection of
messages from twenty different newsgroups with
about one thousand messages in each newsgroup.
We used each newsgroup as the positive class and
pooled the remaining nineteen newsgroups as the
negative class.
Text in the MeSH25 and Reuters databases has
been preprocessed as follows: all alphabetic
characters were lowercased, non-alphanumeric
characters replaced by blanks, and no stemming
was done. Features in the MeSH25 dataset are all
single nonstop terms and all pairs of adjacent
nonstop terms that are not separated by
punctuation. Features in the Reuters database are
single nonstop terms only. Features in the
20Newsgroups are extracted using the Rainbow
toolbox (McCallum 1996).
MedPhrase. We process MEDLINE to extract all
multiword
UMLS®
(http://www.nlm.nih.gov/research/umls/) phrases
that are present in MEDLINE. From the resulting
set of strings, we drop the strings that contain
punctuation marks or stop words. The remaining
strings are normalized (lowercased, redundant
white space is removed) and duplicates are
removed. We denote the resulting set of 315,679
phrases by U phrases .
For each phrase in U phrases , we randomly
sample, as available, up to 5 MEDLINE sentences
containing it. We denote the resulting set of
728,197 MEDLINE sentences by S phrases . From
S phrases we extract all contiguous multiword
expressions that are not present in U phrases . We
call them n-grams, where n>1. N-grams containing
punctuation marks and stop words are removed
and remaining n-grams are normalized and
duplicates are dropped. The result is 8,765,444 ngrams that we refer to as M ngram . We believe that
M ngram contains many high quality biological
phrases. We use U phrases , a known set of high
quality biomedical phrases, as the positive class,
and M ngram as the negative class.
In order to apply machine learning we need to
define features for each n-gram. Given an n-gram
grm that is composed of n
words,
grm  w1w2 wn , we extract a set of 11 numbers
 fi i 1 associated with the n-gram grm. These are
2.3
as follows:
f1: number of occurrences of grm throughout
Medline;
f2: -(number of occurrences of w2…wn not
following w1 in documents that contain grm)/ f1;
f3: -(number of occurrences of w1…wn-1 not
preceding wn in documents that contain grm)/ f1;
f4: number of occurrences of (n+1)-grams of the
form xw1…wn throughout Medline;
f5: number of occurrences of (n+1)-grams of
the form w1…wn x throughout Medline;
A standard way to measure the success of a
classifier is to evaluate its performance on a
collection of documents that have been previously
classified as positive or negative. This is usually
accomplished by randomly dividing up the data
into training and test portions which are separate.
The classifier is then trained on the training
portion, and is tested on test portion. This can be
done in a cross-validation scheme or by randomly
re-sampling train and test portions repeatedly.
We are interested in studying the case where
only some of the positive documents are labeled.
We simulate that situation by taking a portion of
the positive data and including it in the negative
training set. We refer to that subset of positive
documents as tracer data (Tr). The tracer data is
then effectively mislabeled as negative. By
introducing such an artificial supplement to the
negative training set we are not only certain that
the negative set contains mislabeled positive
examples, but we know exactly which ones they
are. Our goal is to automatically identify these
mislabeled documents in the negative set and
knowing their true labels will allow us to measure
how successful we are. Our measurements will be
carried out on the negative class and for this
purpose it is convenient to write the negative class
as composed of true negatives and tracer data
(false negatives)
C'  C  Tr .
11
 p  w1 | w2  1  p w1 | w2   

 1  p  w1 | w2   p w1 | w2  


f6: log 
f7: mutual information between w1 and w2;
 p  wn 1 | wn  1  p  wn 1 | wn   

 1  p  wn 1 | wn   p  wn 1 | wn  


f8: log 
f9: mutual information between wn-1 and wn;
f10: -(number of different multiword expressions
beginning with w1 in Medline);
f11: -(number of different multiword expressions
ending with wn in Medline).
We discretize the numeric values of the
 fi i 1
11
into categorical values.
In addition to these features, for every n-gram
grm, we include the part of speech tags predicted
by the MedPost tagger (Smith, Rindflesch et al.
2004). To obtain the tags for a given n-gram grm
we randomly select a sentence from S phrases
containing grm, tag the sentence, and consider the
tags t0t1t2 tn1tntn1 where t0 is the tag of the
word preceding word w1 in n-gram grm, t1 is the
tag of word w1 in n-gram grm, and so on. We
construct the features

if n  2 :  t0 ,1 ,  t1 , 2  tn ,3 tn 1 , 4  , t2 ,..., tn 1


otherwise:  t0 ,1 ,  t1 , 2  tn ,3 tn 1 , 4 .
These features emphasize the left and right ends of
the n-gram and include parts-of-speech in the
middle without marking their position. The
resulting features are included with
 fi i 1
11
to
represent the n-gram.
158
Experimental Design
When we have trained a classifier, we evaluate
performance by ranking C' and measuring how
well tracer data is moved to the top ranks. The
challenge is that Tr appears in the negative class
and will interact with the training in some way.
2.4
Evaluation
We evaluate performance using Mean Average
Precision (MAP) (Baeza-Yates and Ribeiro-Neto
1999). The mean average precision is the mean
value of the average precisions computed for all
topics in each of the datasets in our study. Average
precision is the average of the precisions at each
rank that contains a true positive document.
3
Results
3.1
MeSH25, Reuters, and 20NewsGroups
We begin by presenting results for the MeSH25
dataset. Table 1 shows the comparison between
Huber and SVM methods. It also compares the
performance of the classifiers with different levels
of tracer data in the negative set. We set aside 50%
of C to be used as tracer data and used the
remaining 50% of C as the positive set for
training. We describe three experiments where we
have different levels of tracer data in the negative
set at training time. These sets are C , C  Tr20 ,
and C  Tr50 representing no tracer data, 20% of
C as tracer data and 50% of C as tracer data,
respectively. The test set C  Tr20 is the same for
all of these experiments. Results indicate that on
average Huber outperforms SVM on these highly
imbalanced datasets. We also observe that
performance of both methods deteriorates with
increasing levels of tracer data.
Table 2 shows the performance of Huber and
SVM methods on negative training sets with tracer
data C  Tr20 and C  Tr50 as in Table 1, but
with cross training. As mentioned in the Methods
section, we first divide each negative training set
into two disjoint pieces Z1 and Z 2 . We then train
documents in the positive training set versus
documents in Z1 to score documents in Z 2 and
train documents in the positive training set versus
documents in Z 2 to score documents in Z1 . We
then merge Z1 and Z 2 as scored sets and report
measurements on the combined ranked set of
documents. Comparing with Table 1, we see a
significant improvement in the MAP when using
cross training.
Table 1: MAP scores trained with three levels of tracer data introduced to the negative training set.
No Cross Training
MeSH Terms
celiac disease
lactose intolerance
myasthenia gravis
carotid stenosis
diabetes mellitus
rats, wistar
myocardial infarction
blood platelets
serotonin
state medicine
urinary bladder
drosophila melanogaster
tryptophan
laparotomy
crowns
streptococcus mutans
infectious mononucleosis
blood banks
humeral fractures
tuberculosis, lymph node
mentors
tooth discoloration
pentazocine
hepatitis e
genes, p16
Avg
No Tracer Data
Huber
SVM
0.694
0.677
0.632
0.635
0.779
0.752
0.466
0.419
0.181
0.181
0.241
0.201
0.617
0.575
0.509
0.498
0.514
0.523
0.158
0.164
0.366
0.379
0.553
0.503
0.487
0.480
0.186
0.173
0.520
0.497
0.795
0.738
0.622
0.614
0.283
0.266
0.526
0.495
0.385
0.397
0.416
0.420
0.499
0.499
0.710
0.716
0.858
0.862
0.278
0.313
0.491
0.479
159
Tr20 in training
Huber
SVM
0.466
0.484
0.263
0.234
0.632
0.602
0.270
0.245
0.160
0.129
0.217
0.168
0.580
0.537
0.453
0.427
0.462
0.432
0.146
0.134
0.312
0.285
0.383
0.377
0.410
0.376
0.138
0.101
0.380
0.365
0.306
0.362
0.489
0.476
0.170
0.153
0.315
0.307
0.270
0.239
0.268
0.215
0.248
0.215
0.351
0.264
0.288
0.393
0.041
0.067
0.321
0.303
Tr50 in training
Huber
SVM
0.472
0.373
0.266
0.223
0.562
0.502
0.262
0.186
0.155
0.102
0.217
0.081
0.567
0.487
0.425
0.342
0.441
0.332
0.150
0.092
0.285
0.219
0.375
0.288
0.402
0.328
0.136
0.066
0.376
0.305
0.218
0.306
0.487
0.376
0.168
0.115
0.289
0.193
0.214
0.159
0.257
0.137
0.199
0.151
0.380
0.272
0.194
0.271
0.072
0.058
0.303
0.238
Table 2: MAP scores for Huber and SVM trained with two levels of tracer data introduced to the
negative training set using cross training technique.
2-fold Cross Training
MeSH Terms
celiac disease
lactose intolerance
myasthenia gravis
carotid stenosis
diabetes mellitus
rats, wistar
myocardial infarction
blood platelets
serotonin
state medicine
urinary bladder
drosophila melanogaster
tryptophan
laparotomy
crowns
streptococcus mutans
infectious mononucleosis
blood banks
humeral fractures
tuberculosis, lymph node
mentors
tooth discoloration
pentazocine
hepatitis e
genes, p16
Avg
Tr20 in training
Huber
SVM
0.550
0.552
0.415
0.426
0.652
0.643
0.262
0.269
0.148
0.147
0.212
0.186
0.565
0.556
0.432
0.435
0.435
0.447
0.135
0.136
0.295
0.305
0.426
0.411
0.405
0.399
0.141
0.128
0.375
0.376
0.477
0.517
0.519
0.514
0.174
0.169
0.335
0.335
0.270
0.259
0.284
0.278
0.207
0.225
0.474
0.515
0.474
0.499
0.102
0.101
0.350
0.353
We performed similar experiments with the
Reuters and 20NewsGroups datasets, where 20%
and 50% of the good set is used as tracer data. We
report MAP scores for these datasets in Tables 3
and 4.
3.2
Identifying high quality biomedical
phrases in the MEDLINE Database
We illustrate our findings with a real example
of detecting high quality biomedical phrases
among M ngram , a large collection of multiword
Tr50 in training
Huber
SVM
0.534
0.521
0.382
0.393
0.623
0.631
0.241
0.241
0.144
0.122
0.209
0.175
0.553
0.544
0.408
0.426
0.417
0.437
0.133
0.132
0.278
0.280
0.383
0.404
0.390
0.391
0.136
0.126
0.355
0.353
0.448
0.445
0.496
0.491
0.168
0.157
0.278
0.293
0.262
0.244
0.275
0.265
0.209
0.194
0.495
0.475
0.482
0.478
0.083
0.093
0.335
0.332
Table 3: MAP scores for Huber and SVM
trained with 20% and 50% tracer data introduced to
the negative training set for Reuters dataset.
Reuters
No CT
2-Fold CT
Tr20 in training
Huber SVM
0.478 0.451
0.662 0.654
Tr50 in training
Huber SVM
0.429
0.403
0.565
0.555
Table 4: MAP scores for Huber and SVM
trained with 20% and 50% tracer data introduced to
the negative training set for 20NewsGroups dataset.
expressions from Medline. We believe that M ngram
contains many high quality biomedical phrases.
These examples are the counterpart of the
mislabeled positive examples (tracer data) in the
previous tests.
160
20News
Groups
No CT
2-Fold CT
Tr20 in training
Huber
SVM
0.492 0.436
0.588 0.595
Tr50 in training
Huber SVM
0.405
0.350
0.502
0.512
To identify these examples, we learn the
difference between the phrases in U phrases and
M ngram . Based on the training we rank the n-grams
berberis aristata, dna hybridization, subcellular
distribution, acetylacetoin synthase, etc. One falsepositive example in that list was congestive heart.
in M ngram . We expect the n-grams that cannot be
separated from UMLS phrases are high quality
biomedical phrases. In our experiments, we
perform 3-fold cross validation for training and
testing. This insures we obtain any possible benefit
from cross training. The results shown in figure 1
are MAP values for these 3 folds.
Figure 1. Huber, CS-SVM, and naïve Bayes
classifiers applied to the MedPhrase dataset.
Comparison of different Machine Learning Methods
Average Precision
0.28
0.26
0.24
0.22
Huber
CS-SVM
Bayes
0.20
1
11
21
31
Cost Factor r+
41
We trained naïve Bayes, Huber, and CS-SVM
with a range of different cost factors. The results
are presented in Figure 1. We observe that the
Huber classifier performs better than naïve Bayes.
CS-SVM with the cost factor of 1 (standard SVM)
is quite ineffective. As we increase the cost factor,
the performance of CS-SVM improves until it is
comparable to Huber. We believe that the quality
of ranking is better when the separation of U phrases
from M ngram is better.
Because we have no tracer data we have no
direct way to evaluate the ranking of M ngram .
However, we selected a random set of 100 n-grams
from M ngram , which score as high as top-scoring
10% of phrases in U phrases . Two reviewers
manually reviewed that list and identified that 99
of these 100 n-grams were high quality biomedical
phrases. Examples are: aminoshikimate pathway,
161
4
Discussion
We observed that the Huber classifier performs
better than SVM on imbalanced data with no cross
training (see appendix). The improvement of
Huber over SVM becomes more marked as the
percentage of tracer data in the negative training
set is increased. However, the results also show
that cross training, using either SVM or Huber
(which are essentially equivalent), is better than
using Huber without cross training. This is
demonstrated in our experiments using the tracer
data. The results are consistent over the range of
different data sets. We expect cross training to
have benefit in actual applications.
Where does cost-sensitive learning fit into this
picture? We tested cost-sensitive learning on all of
our corpora using the tracer data. We observed
small and inconsistent improvements (data not
shown). The optimal cost factor varied markedly
between cases in the same corpus. We could not
conclude this was a useful approach and instead
saw better results simply using Huber. This
conclusion is consistent with (Zhang and Iyengar
2002) which recommend using a quadratic loss
function. It is also consistent with results reported
in (Lewis, Yang et al. 2004) where CS-SVM is
compared with SVM on multiple imbalanced text
classification problems and no benefit is seen using
CS-SVM. Others have reported a benefit with CSSVM (Abkani, Kwek et al. 2004; Eitrich and Lang
2005). However, their datasets involve relatively
few features and we believe this is an important
aspect where cost-sensitive learning has proven
effective. We hypothesize that this is the case
because with few features the positive data is more
likely to be duplicated in the negative set. In our
case, the MedPhrase dataset involves relatively
few features (410) and indeed we see a dramatic
improvement of CS-SVM over SVM.
One approach to dealing with imbalanced data
is the artificial generation of positive examples as
seen with the SMOTE algorithm (Chawla, Bowyer
et al. 2002). We did not try this method and do not
know if this approach would be beneficial for
textual data or data with many features. This is an
area for possible future research.
Effective methods for leveraging positively
labeled data have several potential applications:
 Given a set of documents discussing a
particular gene, one may be interested in
finding other documents that talk about the
same gene but use an alternate form of the
gene name.
 Given a set of documents that are indexed with
a particular MeSH term, one may want to find
new documents that are candidates for being
indexed with the same MeSH term.
 Given a set of papers that describe a particular
disease, one may be interested in other
diseases that exhibit a similar set of symptoms.
 One may identify incorrectly tagged web
pages.
These methods can address both removing
incorrect labels and adding correct ones.
5
h( z )  0 and all C points yield
z  1 and h( z )  2 . The change of the loss function
h( z ) in (2) with a change w is given by
yield z  1 and
dh( z )
 w z  w   yi xi  w (5).
dz
In order to reduce the loss at a C data point ( xi , yi ) ,
h  z  
we must choose
w such that xi  w  0. But we
assume that there are significantly more C class
data points than C and many such points x are
xi such that x  w  0.
Then h( z ) is likely be increased by x  w( 0)
mislabeled and close to
for these mislabeled points. Clearly, if there are
significantly more C class data than those of C
class and the C set contains a lot of mislabeled
Conclusions
Given a large set of documents and a small set
of positively labeled examples, we study how best
to use this information in finding additional
positive examples. We examine the SVM and
Huber classifiers and conclude that the Huber
classifier provides an advantage over the SVM
classifier on such imbalanced data. We introduce a
technique which we term cross training. When this
technique is applied we find that the SVM and
Huber classifiers are essentially equivalent and
superior to applying either method without cross
training. We confirm this on three different
corpora. We also analyze an example where costsensitive learning is effective. We hypothesize that
with datasets having few features, cost-sensitive
learning can be beneficial and comparable to using
the Huber classifier.
Appendix: Why Huber Loss Function works
better for problems with Unbalanced Class
Distributions.
The drawback of the standard SVM for the
problem with an unbalanced class distribution
results from the shape of h( z ) in (2). Consider the
initial condition at w  0 and also imagine that there is
a lot more
this case, by choosing   1 , we can achieve the
minimum value of the loss function in (1) for the initial
condition w  0 . Under these conditions, all C points
C training data than C training data. In
162
points, it may be difficult to find w that can
result in a net effect of decreasing the right hand
side of (2). The above analysis shows why the
standard support vector machine formulation in (2)
is vulnerable to an unbalanced and noisy training
data set. The problem is clearly caused by the fact
that the SVM loss function h( z ) in (2) has a
constant slope for z  1 . In order to alleviate this
problem, Zhang and Iyengar (2002) proposed the
loss function h 2 ( z ) which is a smooth nonincreasing function with slope 0 at z  1 . This
allows the loss to decrease while the positive
points move a small distance away from the bulk
of the negative points and take mislabeled points
with them. The same argument applies to the
Huber loss function defined in (4).
Acknowledgments
This research was supported by the Intramural Research
Program of the NIH, National Library of Medicine.
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